Dynamical Critical Behaviors of the Ising Spin Chain: Swendsen-Wang and Wolff Algorithms

TL;DR

This paper analytically investigates the zero-temperature Ising chain's dynamical critical behavior under Swendsen-Wang and Wolff algorithms, revealing scaling laws and domain length distributions.

Contribution

It provides the first analytical derivation of domain length distributions and scaling properties for the Ising chain under these cluster algorithms.

Findings

Density of unreacted domains scales as <l>^{-3/2}

Derived domain length distribution for Swendsen-Wang dynamics

Computed domain length distribution for Wolff dynamics

Abstract

We study the zero-temperature Ising chain evolving according to the Swendsen-Wang dynamics. We determine analytically the domain length distribution and various ``historical'' characteristics, e.g., the density of unreacted domains is shown to scale with the average domain length as <l>^{-d} with d=3/2 (for the q-state Potts model, d=1+1/q). We also compute the domain length distribution for the Ising chain endowed with the zero-temperature Wolff dynamics.